Partial preview of the text
Download 3d geometry notes pdf and more Summaries Geometry in PDF only on Docsity!
E M * 8D Geometry. Slwagatet Lines 1. Fquarkton of. a chraig het Ahrovep Vw i Aine passin port ay na cael to ow ane vector: , y @NVoclos form’ us ate Bop [t = porameter} + e Cartesian jorme 5 A=(% Dy) K-% _ Y-y z-4 eae = = “1. t.=t B= Ata my tok = a e Non- poramet se Vector Cay.” : (#-a)x b. & 8; D., Cquatiow oh a Gywai oft line in through wo epee i adel a B Voter Jorn: v= Q+t(b-a) Pe * Comtestan formu? Ry YY @ = 2} A= (21.91%) Xy- A eres ~ Z —2i° = Coo Yo1 22). nN: e Sayre Zk ° Non-porometyte Neactor yu. : R= KY (#- 2) * (b-2) = 9 ¥ Angle ®Melwoon “Two lames! @Veclor oom t= Ziv , = T+ta Bea B = Cos7 7 Ie] al Sayochich Saha e Comlestan Loran + KH YB zeB , MeL eH Ko %y = Jos - 9 a b, cy a, bj cy 2 2 a Day -+ bribe £orGe @ = con! N74 b74 0,7 Al 4,?+4b,%4 6,” § Qhow Limes? “Two Wines which ave [= hether wrlerec Ary now -- PRE cpawothelL ase called Skow Imor. d ~p t * Showetest distance between lwo Aineas? Shosles+ déctance between +wo MAecsectng Agass FS Aer, _ _ ~_ 2. Pasalle? Limec? P= a, ++b = — ¥E CTPA + 4b = es a= | bx Ca,- 4) | lvl i 3, Skow Lanes! F- At tbr Daa =? re m= Att a nus ya, Wea) Cia SOI > }b,* b,| ° mn % y-y Z-z dos4 oTme 0 e 1 oe i} e Cartesian, form Ay: a, “ey a 4, > R-%_ Y-y2 _ 27 Fe a, bz o 2a YM 2-2 Ay by ont d= a2 b3 cD 2 { Cyg-br9)+C, a - Oy) 4 (a,b, - a) Sayochich Saha 2, Equactton of. -the. kane when otks-lance from the erie anc init hosmart %¢ i & _>, ‘ © Vocclowr Lorm 2 YT. = P: Ee — unit nowmal P ~ dis-lance | : © Carlesiaw form: A s ly i we Ay ray + 2K Aw + ny +N = Pp: Si, Eau 6f, the plane parsing. throughs a ane" porwt hu paradbet -to -lwo vectore? om > © Veet r fowm - y= O4+tD+t uc _" L a — Aran point parallet to BAe] e Cavrtestan lorm? ‘ a a“ . a= yi+y,y +2* Aa Bey Yay; 2-2 ~~ > iN A ae H kK 4 Im, hy = 0 b= Alamy + hy = € o a a A, mo Ry = Agi army t mn S Aen paromedete, term? [Gra B 2] =o alee ey -Laee) 4. Pq uation Dh the plane passing Urvoup'v ats given pots RE i cies to ow el vector? e Vector form * => tu Veeteos of pont, parottiel +o v] . o Non — -poranusiete : L@- a) (6-7) vy =O, Sayochich Saha 3D oery) 3 e % * a Cowlosiaw form ae x,t 49,5 4 Z,h i - — 4 oa A me ty Yo sti 7-7) De td ZK Ky— Hy = %-%| = 0. A o + Ba Aha mye evK: Xx ms w 5. Equattow oh the plane — pacerrg. through Shree given nou— cotlineav punts : © Vector for: ns is a+ ¢(b- a) + t(e-2 yao [ 0,6,0 — pasion Vectsrs of pan « Non poramertric ‘ [c#-%) (2-3) (@-7)]=0. ¢ Caglesiany ferm! HX Yd, Z- 2, %y-% YoY F2- 2h = 0, Ky~ Ry J3-4 Zy- 4 6. Equattow of a plane Passing Anroug hy the Aine al. wrlers2cttow aL two sia pldves | © Vector form qu af plane passing Thaoug & the = Ale ob aprlersect tow of epboies, wo ny = G, a x. hy =4¢, nay (#4 - ht a (A - 4) = 0. OA,xt+ by +c ata, = 1°] =O. O,% + bsy +o. 2+ 42 = =0. An+ b Cc aj) +r (a x%+ b, c,ztay (aie bieeay e+ ly) OA Ra ey ? Gage Saba 0 Carlesiaw fet: 3D Gooenerby ; ¥ Jirlercad-l Lorn of the agu™ af a plan ! /-Loeg, O,b,0 9 KN, QXes inlorcafrt q Coblonarty of dwo Wnast > _ iid Lames. = Opit “ by R= a,+’b, itt = Concttan (@-a,). (Av) Ae MX Ya Fa a, b, far =0, Qa b, SS + Angke between ~-lwo flanes : wen, = tN p= cost [ RP nn oS Mong = Vy 4 Ange between @& AN and a tance: We atte oh Pele es [ bn | Ton = YY, x Pqporton ob bisactoss of the Ongte hortwcow -wo plones: Ax+ by + eZ#+aA=0 Tad, do} Qua + ay +Qz7+d,=0. i Oxtbyacet+ A Onn + by Ope Voreee ake Ba Ge GO, + bb, +c, Acute angh. Obtuse anghe bisects btsector >0 _ e “ . Tayachich Saha s Jimage- of Q@ rpornt wtth weRspeet +o we pana aad Jirmnage of A(x, Nv #1) wry Ant byt ce+d= 0 be ® (x2 54.0%) KH Jae % Z,- 4, —- 2 (ax % + = = + by, +0: od a a z a = (ax, Uy FOR gk p) a+b +6, 4 Poot of perpendicutars _ -~(QX, +by, 462, +4) & b Cis Qg?+ b> +<%, y—% Ya Poni 24 + Rol, Jecttow a @ 4 lane ow another clone: “The aot kel of the tanec an+ by + cz+ A =O ae the 1 Qe Q;x+ bry + Cjz+ ad, =0 tS, 2(aq, +b) + a) (xt by + oztay) = (aP4br4 ay) (ar+by +cz+d) # An oxpsasttaws of — seconcl one a (hemogenzour) te Ppwe gata tao planes assang. through origin. Concdsttow — a h 4 hn b f <= O : 4 fie x of 8 of the acude anghe between two planes avy by ™+ azt4 2fyet 24 2% + Dhay > fo) then os ghee {aus th*— be~ (ea~ar) GQt+ht+e f. a planes ota ? aban ALeuhar, +hen Atb+e = O. Sayochich Saha 
